A Diagrammatic Kinetic Theory of Density Fluctuations in Simple Liquids
in the Overdamped Limit. I. A Long Time Scale Theory for High Density

Abstract

Starting with a formally exact diagrammatic kinetic theory for the equilibrium correlation functions of particle density and current fluctuations for a monatomic liquid, we develop a theory for high density liquids whose interatomic potential has a strongly repulsive short ranged part. We assume that interparticle collisions via this short ranged part of the potential are sufficient to randomize the velocities of the particles on a very small time scale compared with the fundamental time scale defined as the particle diameter divided by the mean thermal velocity. When this is the case, the graphical theory suggests that both the particle current correlation functions and the memory function of the particle density correlation function evolve on two distinct time scales, the very short time scale just mentioned and another that is much longer than the fundamental time scale. The diagrams that describe the motion on each of these time scales are identified. When the two time scales are very different, a dramatic simplification of the diagrammatic theory at long times takes place. We identify an irreducible memory function and a more basic function, which we call the irreducible memory kernel. This latter function evolves on the longer time scale only and determines the time dependence of the density and current correlation functions of interest at long times. In the following paper, a simple one-loop approximation for the irreducible memory kernel is used to calculate correlation functions for a Lennard-Jones fluid at high density and a variety of temperatures.

pacs:

Equilibrium time correlation functions of particle density and momentum density are important to the kinetic theory of liquidsBoonYip (); McQuarrieBook (); HansenMacDonald (); MazenkoBook () because their long time behavior can be related to various bulk transport coefficients and because they can be measured in scattering experiments and easily calculated in molecular dynamics simulations, making them useful in determining the accuracy of a theory that can calculate them.

Time correlation functions of dynamical variables can be expressed as the solutions of formally exact integro-differential equations that typically contain a term in which the correlation function is convoluted in time with its memory function.HMori () If the memory function is known, the correlation function can be computed easily, but the memory function is generally too complicated to be evaluated exactly. Consequently, much theoretical effort has been expended throughout the last several decades to come up with tractable approximation schemes for memory functions that lead to reasonable results for their corresponding correlation functions.

Many correlation functions of interest contain the effects of physical processes that take place on different time scales. This separation of time scales may allow one to express the memory function as the sum of a short time component and long time component and make approximations to the two components separately.

One of the most fruitful applications of this general strategy is the mode coupling theory of Götze and coworkers,Gotze1 (); LesHouches (); GotzeSjogren1 (); GotzeSjogren2 (); GotzeSjogren3 (); GotzeSjogren4 () who were interested in constructing a long time scale theory of a two-point density correlation function. They approximated the short time part of its memory function as a function proportional to a Dirac delta function of time. They approximated the long time part of the memory function as a four-point density correlation function that was then further approximated self consistently as a product of two two-point density correlation functions.

SjögrenSjogren1 (); Sjogren2 (); Sjogren3 () also developed a kinetic theory that hinged on separating the memory function into a short and long time part. Using the fully renormalized kinetic theory of MazenkoFRT1 (); FRT2 (); FRT3 (); MazenkoYip () as a starting point, Sjögren assumed that the short time scale dynamics were dominated by binary collisions and that the long time scale dynamics consisted of recollisions and hydrodynamic backflow and used these assumptions as the basis for his separation. This had the advantage of allowing him to derive formal expressions for both the short and long time parts of his memory function, but the cost of trying to describe the dynamics on all time scales simultaneously was that these expressions were extremely cumbersome and difficult to evaluate.

AndersenDKT1 (); DKT2 (); DKT3 () much later developed a formally exact diagrammatic kinetic theory for correlation functions of phase space densities, some of whose features and results were closely related to those of Mazenko’s fully renormalized kinetic theory. The diagrammatic theory had the advantage of being able to represent both correlation functions and their memory functions in a unified way in terms of diagrams that could be ascribed straightforward physical interpretations thanks to the individual elements of the diagrams being relatively simple (even though the full diagrams in general were not). Andersen and coworkersDKT4 (); DKT5 (); DKT6 () then used the physical intuition this theory afforded to derive several candidate graphical representations of the short time part of the memory function of the phase space density correlation function that could be evaluated numerically and used to compute the short time behavior of several correlation functions of interest. Each of these representations in turn implied an explicit diagrammatic series for the corresponding long time part of the memory function, but these series were always too complex to evaluate.

In this paper we develop a new method for characterizing the multiple time scale behavior of simple atomic liquids that are in what we call the ‘overdamped limit’. For the following qualitative discussion, let us regard the unit of time for an atomic liquid to be the ratio of the particle diameter and the mean thermal velocity and express all times in terms of that unit. Let ν be the average rate at which a particle in the fluid experiences uncorrelated binary repulsive collisions. Then ν−1 is approximately the relaxation time for fluctuations in a particle’s momentum at equilibrium. The overdamped limit is defined by the assumption that the short ranged repulsive forces of the particles are hard enough and the density is high enough that uncorrelated binary repulsive collisions randomize the velocities of the particles on a time scale much smaller than 1. In the overdamped limit, ν is large compared with 1, and ν−1 is small compared with 1.

We start from the exact diagrammatic theory for the correlation function of phase space density fluctuations and focus on the diagrammatic series for a projected propagator associated with its memory function. After a number of renormalizations, one of which makes use of a short time theory equivalent to the generalized Enskog theory for hard spheres, we obtain a graphical formulation for the projected propagator that shows that its magnitude and time dependence is of the form

χP(t)≈(constant)exp(−νt)+ν−2h(t/ν,ν−1),

where the function h has only nonnegative powers of its two arguments. The two terms vary on two very distinct time scales. The first term decays very rapidly to 0, on a time scale of O(ν−1). This is the time scale for randomization of the momentum of a particle by uncorrelated binary repulsive (or hard sphere) collisions. The second term varies on a time scale of O(ν). This is the time scale for a particle to diffuse a distance equal to its diameter when its self diffusion coefficient is O(ν−1), which is the value determined by the repulsive collisions. If ν is large and t≫ν−1 , it is reasonable to approximate the behavior of the projected propagator as ν−2h(t/ν,0). In the diagrammatic theory, this corresponds to retaining only a small subset of the diagrams in the original series for the projected propagator. These diagrams are characterized topologically, and from that point on, the theory works only with the diagrams that vary on the long time scale of O(ν) and that are lowest order in powers of ν−1.

We note that SzamelSzamel2 () has derived a diagrammatic kinetic theory for Brownian particles undergoing diffusive motion in a solvent, using methods similar to those used in deriving the diagrammatic kinetic theory for particle systems undergoing Hamiltonian dynamics.DKT1 (); DKT2 (); DKT3 () Szamel’s results and the present results have several parallels. However, it should be noted that Szamel’s method starts from the theory of Smoluchowski dynamics in a solvent whose microscopic dynamics is not explicitly discussed, whereas the present work starts from Hamiltonian dynamics for a one component particle system and shows how the longer diffusive time scale arises from a mechanical description that includes all the shorter time scales and all the degrees of freedom of the system.

In Sec. II we state some of the results of the diagrammatic kinetic theory of time correlation functions of the density in single particle phase space for a monatomic fluid. In Sec. III, we express the potential of mean force as the sum of a short ranged repulsive part and a longer ranged part, approximate the short ranged repulsive part as a hard sphere potential, and use this to express each interaction vertex in the graphical theory as a sum of two corresponding parts. In Sec. IV, we introduce a Hermite polynomial representation of the momentum dependence of diagrams. Sec. V develops the kinetic theory in the Hermite polynomial representation, defines a ‘projected propagator’ χP in diagrammatic terms, and shows that this propagator is directly related to observable correlation functions of interest. Sec. VI defines χEP, the generalized Enskog approximation for the projected propagator, and presents a diagrammatic expression for χP in terms of χEP. Sec. VII discusses the behavior of the diagrams in χP in the overdamped limit and identifies the diagrams that are most important in that limit. It also presents an especially useful diagrammatic representation of χP in terms of an ‘irreducible memory kernel,’ mirr, as well as a diagrammatic series for mirr that describes its slowly relaxing behavior. (Appendix E gives the detailed relationships that allow the correlation functions of interest to be calculated from the irreducible memory kernel.) Sec. VIII closes with a discussion. In a subsequent paper,paper2 () a one-loop approximation for the irreducible memory kernel is formulated and evaluated for a dense Lennard-Jones fluid, and the correlation functions that follow from that result are compared with molecular dynamics computer simulation results.

ii.1 Definitions

The system of interest is a one component classical fluid of identical point particles at equilibrium. We consider a canonical ensemble of such systems with N particles in volume V with temperature T. The positions and momenta of the particles are rN={r1,…,rN} and pN={p1,…,pN}. We define a density in single particle phase space as

(1)

where δ denotes a Dirac delta function. The fluctuation of this density from its canonical ensemble average value is

δf(R,P,t)≡f(R,P,t)−⟨f(R,P,t)⟩,

(2)

where the angular brackets denote an ensemble average. This is a dynamical variable whose value fluctuates as the particles’ coordinates and momenta change with time. The time correlation function of this dynamical variable

C(R,P,t;R′,P′,t′)≡⟨δf(R,P,t)δf(R′,P′,t′)⟩

is a fundamental function of interest in the kinetic theory of gases and liquids.FRT1 (); FRT2 (); FRT3 (); MazenkoYip (); Sjogren1 (); Sjogren2 (); Sjogren3 (); BoonYip (); MazenkoBook () It is closely related to the time correlation functions of particle density and momentum density, which are related to transport properties and to the observables in coherent neutron scattering experiments. A closely related function is the corresponding self correlation function.

Cs(R,P,t;R′,P′,t′)≡N∑i=1⟨δfi(R,P,t)δfi(R′,P′,t′)⟩

Here

fi(R,P,t)≡δ(R−ri(t))δ(P−pi(t))

Cs is related to the velocity autocorrelation function and to incoherent neutron scattering.

For t=t′ these functions are equilibrium static correlation functions.

C(R,P,t;R′,P′,t)=F1(R,P;R′,P′)

=

ρMM(P)δ(R−R′)δ(P−P′)

+ρ2MM(P)MM(P′)(g(R−R′)−1)

=

ρMM(P)δ(R−R′)δ(P−P′)

Here MM is the Maxwell-Boltzmann distribution of momentum, and g is the usual pair correlation function.

ii.2 Diagrammatic theory of time correlation functions

In previous papers,DKT1 (); DKT2 (); DKT3 (); DKT4 (); DKT5 (); DKT6 () we have developed a diagrammatic theory of a hierarchy of time correlation functions. The theory defines retarded propagators χ and χs for the correlation functions C and Cs, respectively,Note1 () and expresses the C function for positive values of t−t′ in terms of the t=t′ value in the following way.

C(R,P,t;R′,P′,t′)

=

∫dR′′dP′′χ(R,P,t;R′′,P′′,t′)

×F1(R′′,P′′;R′,P′)

=

∫dR′′dP′′χs(R,P,t;R′′,P′′,t′)

×Fs1(R′′,P′′;R′,P′)

The theory gives the following diagrammatic expressions for the propagators.

χ(R,P,t;R′,P′,t′)= the sum of all topologically distinct matrix diagrams with:

(i) a left root labeled (R,P,t) and a right root labeled (R′,P′,t′);

(ii) free points;

(iii)χ(0) bonds;

(iv)Qc111, Qc011, Qc112, Qc121, and Qc222 vertices;

such that:

(i) each root is attached to a bond;

(ii) each free point is attached to a bond and a vertex. □

χs(R,P,t;R′,P′,t′)=
the sum of all diagrams in the series just above that have a particle path from the left root to the right root. □

Fig. 1

Figure 1: Some of the simplest diagrams in the series for χ(R,P,t;R′,P′,t′). Each root point is a small open circle. Each free point is a small closed circle. Each χ(0) bond is a line. Each Qnm vertex is a large circle with n points on the left and m points on the right. Each vertex has zero, one, or two internal lines connecting left and right points on the same vertex. A Qc111 vertex has an internal line, which distinguishes it from a Qc011, which does not. A Qc222 vertex has two internal lines that are not topologically equivalent, so one is drawn as a solid line and the other as a dashed line. (In the actual series, each left root is labeled (R,P,t), and each right root is labeled (R′,P′,t′), but these labels have been deleted from the figure for simplicity.)

illustrates some of the diagrams in the series for χ.
The graphsNOTE2 () in this series are similar to Mayer cluster diagrams for static correlation functions of classical fluids at equilibrium,Mayer1 (); MayerBook (); McQuarrieBook (); HansenMacDonald (); MoritaHiroike (); GStell (); HCA1 () but they differ in several respects. In particular, the left points and right points of a vertex or bond are not equivalent, and, to represent this, a graph should be drawn so that it is clear which points are on the left and which are on the right on every vertex or bond. (If necessary, each bond can be drawn with an arrowhead pointing from its right point to its left point. In the figures in this paper, we adhere to the convention that the left point of a bond is always to the left of the right point, so arrowheads are not needed to make the distinction between the two nonequivalent ends of a bond.) When a free point is attached to a bond and a vertex, it is attached to the left point of the bond and a right point on the vertex, or to the right point of the bond and a left point on the vertex. When a vertex has a left point that is also attached to a bond, we say that ‘the vertex has the bond on the left’, with an analogous meaning to the phrase ‘the vertex has the bond on the right’. Some vertices have internal lines that connect right and left points on the vertex.

The Q vertices describe fundamental dynamical processes, such as free particle motion and particle interactions, that affect the fluctuations of f. The bonds describe the causal time evolution of fluctuations of f. (Time increases from right to left in these diagrams.) Each diagram in the series above represents a single term in an iterative solution of the equations of motion for χ and χs.
Infinite series of diagrams such as those above can be manipulated using the topological reduction techniques developed by Morita and Hiroike.MoritaHiroike () (See also StellGStell () and Andersen.HCA1 ())

Like Mayer diagrams, each diagram has a real numerical value that is a function of the arguments assigned to the root points. Each vertex and bond in a diagram is associated with a function. To evaluate a diagram, dummy variables for position and momentum are assigned to each free point, a dummy time variable is assigned to each vertex, and an expression that contains the bond functions and vertex functions is constructed in a way that is based on the structure of the diagram. This expression is integrated over all values of the dummy variables to give the value of the diagram.

An overall review of the graphical kinetic theory is presented by Ranganathan and Andersen,DKT4 () including the general method for calculating the value of a diagram. A summary of much of the graphical terminology used in the theory is given by Andersen DKT1 () in an appendix. Other background information is given in Appendix A of the present paper. The diagrammatic theory of self functions, like Cs, involves the topological notion of a particle path in a diagram, which is discussed in Appendix A.1 of the present paper. Explicit expressions for the functions associated with the vertices and bonds in these diagrams are given in Appendix A.4 of the present paper.

The fundamental theory that leads to this diagrammatic statement is a formally exact theory for the dynamics of density fluctuations in single particle phase space. A density fluctuation at a point in single particle phase space corresponds physically to the presence or absence of a particle at that point in the space, as should be clear from Eqs. (1) and (2). The physical interpretation can be useful for deciding how to manipulate the graphical series and for devising approximations. (The physical interpretation of diagrams is discussed in more detail by Andersen and Ranganathan.DKT4 ())

The exact diagrammatic theory has many more Q vertices than those given in the expression above. In the present work we have retained the Qc111 vertex, which describes the free particle motion of a density fluctuation associated with a single particle, and the Qc011, Qc121, Qc112 and Qc222 vertices, which describe the interaction of density fluctuations associated with two distinct particles.
The latter take into account the equilibrium structure of the fluid around the fluctuations. Each function associated with these interaction vertices contains the equilibrium pair potential of mean force or the equilibrium direct correlation function. The vertices we neglect are similar to those retained except that they involve equilibrium correlation functions for three or more particles. We suspect that the vertices that are neglected represent minor corrections that do not introduce new physical effects. (The vertices that are included have enough physics to derive the linearized Boltzmann equation for fluctuations in a dilute gas and the linearized generalized Enskog theory for fluctuations of a dense liquid, as well as describe hydrodynamic behavior for small wave vectors.) The theory is fully renormalized, in the sense of Mazenko,FRT1 (); FRT2 (); FRT3 (); MazenkoYip () in that the vertices retained are expressed in terms of equilibrium static correlation functions with no reference to the bare interparticle interaction. The resulting theory has a rich structure that takes into account the time evolution of arbitrarily many density fluctuations. The ultimate test of the theory will come when its predictions are compared with computer simulation results.

The vertices Qc112, Qc222, and Qc121 contain the potential of mean force of the fluid of interest and describe density fluctuations associated with two distinct particles interacting with each other via this potential. (See Appendix A.4.) If the bare potential itself is very repulsive at short distances and the density is high, the potential of mean force will also be very repulsive at short distances. We assume the bare potential satisfies this condition. It is useful to separate the potential of mean force into its short ranged repulsive part and its longer ranged part, in the same way as the bare potential is separated in the WCA theory of equilibrium liquids,WCArepulsiveforces () because of the different physical effects of these two parts on the dynamics of the fluid. (See Appendix B.1 for the details.)

Then each Qc112, Qc121, and Qc222 vertex can be expressed as the sum of a repulsive (R) part and a longer ranged (L) part, and χ can be expressed in terms of these new vertices, whose functions are given in Appendix B.1.

Each of the functions defined graphically in this paper has a self version. At this point, the verbal description for graphs in the self part will begin to become lengthy and detailed, despite the fact that the actual analysis is straightforward, so henceforth we will present a graphical expression for the self part of a function only if it will appear in the final results.

The QR vertices represent interparticle interactions that are generated by the repulsive part of the potential of mean force. It is useful to consider the effect of completed binary collisions generated by this repulsive potential. To do this, we start by defining a new set of vertices called MR, TR12, TR21, and TR22, that can be used to replace the QR vertices. Each new vertex represents a binary collision between two particles as a result of their short ranged repulsive forces. (MR is a memory function for repulsive collisions, and the TR vertices are scattering functions for repulsive collisions.) The procedure for constructing these vertices is discussed in Appendix B.2.

We are interested in interatomic potential functions for which the short ranged repulsive forces are very strong and in time scales that are long compared with the time required for the completion of a short ranged repulsive force collision of two particles. We want to approximate the MR and TR vertices by the corresponding vertices for hard spheres. The procedure for doing this is discussed in Appendix B.4. The net result is that the MR and TR vertices are replaced by MH and TH vertices, where the H denotes that they are hard sphere vertices. Expressions for the functions associated with these vertices are given in Appendix B.6.

For the development of the present theory, it is worthwhile to expand the propagator in a special complete basis set of Hermite polynomial functions of a momentum variable. Such Hermite polynomial representations of kinetic theories have been used many times.Sjogren2 (); MazenkoWeiYip (); MazenkoYip2 (); DKT4 (); DKT5 ()

χ(R,P,t;R′,P′,t′)

(3)

=

∑λλ′χ(R,λ,t;R′,λ′,t′)

×hλ(P)hλ′(P′)MM(P)MM(P′)

The special Hermite polynomial hλ(P) is defined as follows.

hλ(P)≡

[2λx+λy+λzλx!λy!λz!]−1/2

×Hλx(Px)Hλy(Py)Hλz(Pz)

(4)

The index λ is an ordered triplet of three nonnegative integers (λx,λy,λz), which we refer to as a Hermite index. Here, the momentum P is equal to (2mkBT)−1/2P, and the functions Hn(x), for nonnegative integers n, are Hermite polynomials.

We shall refer to χ(R,λ,t;R′,λ′,t′) as a Hermite matrix element. Each such matrix element can be represented as an integral containing χ(R,P,t;R′,P′,t′) by using the orthonormality relationship for the basis functions.
Each vertex function and bond function in the theory can be similarly expressed in terms of matrix elements. (See Appendix C for the details.) When it is unambiguous to do so, we will abbreviate the triplet of a Hermite index by omitting the commas, so, for example, we will write (001) and (000) rather than (0,0,1) and (0,0,0). We also use the notation ^0≡(000), ^x≡(100), ^y≡(010), and ^z≡(001).

In this Hermite representation, the diagrammatic expression for χ is the same as in the momentum representation at the end of Section III except for the replacement of momenta by Hermite indices. This diagrammatic expression is given just below in Sec. V.1.

When a diagram is evaluated, the matrix elements of the bond and vertex functions appear as factors in an expression that is summed and integrated over dummy variables assigned to the free points, in a way very similar to the evaluation process in the momentum representation. See Appendix C.2 for a detailed statement of the rules for evaluating a diagram and an example.

For simplicity, we adopt a notation that uses the same symbol for the Hermite matrix element of a function as for the function itself, relying on the arguments to distinguish the two functions. (For the remainder of the paper, unless momentum arguments are explicitly indicated, the Hermite matrix representation is to be understood as being used.)

All the bonds in the theory and many of the vertices and other functions of interest are represented by graphs with one left root and one right root, and so the corresponding functions have one left Hermite index and one right Hermite index. For such functions we occasionally use a notation in which the indices are written as subscripts. For example, a function of the form G(R,λ,t;R′,λ′,t′) could be written as G(R,t;R′,t′)λλ′. This is especially useful since many of the quantities calculated involve matrix products of these functions.

All functions of this form that appear in the theory are translationally invariant, stationary, and retarded in time (i.e. the function is defined for all values of its time arguments, but the function is zero if the left time argument is earlier than the right time argument). Thus spatial Fourier transforms and Laplace transforms with regard to time can be useful, especially since many relationships among the functions are convolutions in space and/or time. We use the following definition of the Fourier transform and Laplace transform.

^G(q,t)λλ′≡1V∫VdRdR′e−iq⋅(R−R′)G(R,t;R′,0)λλ′

(5)

~G(R,z;R′)λλ′

≡

∫∞0dte−ztG(R,t;R′,0)λλ′

^~G(q,z)λλ′

≡

∫∞0dte−zt^G(q,t)λλ′

with an analogous notation for Fourier transforms of functions of position only and Laplace transforms of functions of time only.

Relationship of the propagator to observables

Some of the observable quantities of interest in the kinetic theory are the coherent intermediate scattering function, the longitudinal current correlation function, the transverse current correlation function, the incoherent intermediate scattering function, and the incoherent longitudinal current correlation function, which are all functions of wave vector and time. The last of these, for zero wave vector, is proportional to the velocity autocorrelation function. For definitions of these quantities, see, for example, Boon and Yip.BoonYip ()

The time dependence of these functions is closely related to the time dependence of the Fourier transform of some of the matrix elements of χ. It is straightforward to show that

^ϕρ(q,t)

=

^χ(q^k,t)^0^0

(6)

^ϕjl(q,t)

=

^χ(q^k,t)^z^z

(7)

^ϕjt(q,t)

=

^χ(q^k,t)^x^x

(8)

^ϕρs(q,t)

=

^χs(q^k,t)^0^0

(9)

^ϕjls(q,t)

=

^χs(q^k,t)^z^z

(10)

where the ϕ functions are the five correlation functions mentioned above, respectively, when normalized to be unity at zero time, and ^k is the unit vector in the z direction.

Another expression for the propagator

The Hermite matrix elements of χ(0) are in Eq. (37).
We decompose χ(0) into two parts.
Let

χ(0)(R,λ,t;R′,λ′,t′)

=

χ(0)^0(R,λ,t;R′,λ′,t′)+χ(0)P(R,λ,t;R′,λ′,t′)

where

χ(0)^0(R,λ,t;R′,λ′,t′)=χ(0)(R,λ,t;R′,λ′,t′)δ(λ,^0)

χ(0)P(R,λ,t;R′,λ′,t′)=χ(0)(R,λ,t;R′,λ′,t′)[1−δ(λ,^0)]

Both of these are diagonal in their Hermite indices. The χ(0)^0 bond function has just one nonzero element, for λ=λ′=^0.

We can use this decomposition to replace the χ(0) bond by the sum of its two parts, thereby giving a series with χ(0)^0 and χ(0)P bonds. A Qc matrix element all of whose Hermite indices are ^0 is equal to zero. (This is a property shared by all Qc vertices that can appear in this series for χ. See Appendix C.3 for explicit statements of the matrix elements.) If a Qc vertex in a diagram has only χ(0)^0 bonds attached, the diagram will then have a value of zero. Thus it is permissible to include the requirement that each Qc vertex is attached to at least one χ(0)P bond without changing the value of the series. This leads to the following graphical statement.

χ(R,λ,t;R′,λ′,t′)≡ the sum of all topologically distinct matrix diagrams with:

The relationship between the propagator and the coherent intermediate scattering function

To calculate the coherent intermediate scattering function using Eq. (6), we need an expression for the ^0^0 matrix element of χ. The series for this matrix element can be obtained by setting λ=λ′=^0 in the expression for χ in Sec. V.1.2.

We now analyze this graphical series as a step toward evaluating the coherent intermediate scattering function. The method of analysis is similar to that previously usedDKT2 () to analyze a graphical series to obtain a memory function and to the analysis of graphical theories of quantum many-body phenomena to extract self-energies.Dyson ()

We examine each graph and find the χ(0)^0 bonds whose removal would disconnect the diagram. Let n be the number of such bonds in a diagram. For one graph, namely the one with a χ(0)^0 bond between the roots, n=1. For all other diagrams, n≥2, and removal of all n bonds disconnects the diagram into two disconnected roots and n−1 disconnected parts, the latter of which are still internally connected. We take each such part and replace the free point on the far left with a left root and the free point on the far right with a right root. Both roots are assigned the Hermite index ^0, because in the original χ diagram the corresponding free points were attached to χ(0)^0 bonds. The sum of all topologically different graphs that can be obtained by this procedure defines a function to be called M(R,t;R′,t′).

No graph that contributes to M can contain only one vertex and no bond. Such a vertex would have been attached to only two bonds in the original diagram and both bonds would be χ(0)^0 bonds. Such diagrams do not appear in the χ series because of the last requirement in the statement of the series. Therefore every diagram in the series for M(R,t;R′,t′) must have two or more vertices and must be a member of the following series.

M(R,t;R′,t′)≡ the sum of all topologically distinct matrix diagrams with:

(i) the left root is attached to a vertex and the right root is attached to a different vertex;

(ii) each free point is attached to a bond and a vertex;

(iii) each Qc vertex is attached to at least one χ(0)P bond;

(iv) there is no χ(0)^0 bond whose removal would disconnect the roots. □

It follows that

χ(R,^0,t;R′,^0,t′)=χ(0)^0(R,^0,t;R′,^0,t′)

+∫dR′′dt′′dR′′′dt′′′χ(0)^0(R,^0,t;R′′,^0,t′′)

×M(R′′,t′′;R′′′,t′′′)χ(R′′′,^0,t′′′;R′,^0,t′)

(The first term on the right is the value of the diagram in the series for χ with only one bond. All the remaining diagrams in χ have a χ(0)^0 bond on the left, followed by a member of the series for M, followed by a member of the series for χ itself. Summing over all the possibilities gives the second term on the right.)
Note that all of the functions on the right that are functions of time are retarded, and this implicitly limits the ranges of time integrations.

Taking the time derivative and Fourier transform of both sides of the equation and setting q=q^k gives

∂^χ(q^k,t)^0^0∂t=δ(t)+∫t0dt′^M(q^k,t−t′)^χ(q^k,t′)^0^0

(11)

This is a memory function equation for the propagator matrix element associated with the coherent intermediate scattering function. Thus the function M can be regarded as the memory function for the coherent intermediate scattering function.

In the graphical series for M, the vertex that is attached to the left root cannot have two left points. Given the way the roots are labeled with ^0 Hermite indices, the only possibility for the vertex on the left is Qc111 because the matrix elements of Qc011, Qc1L12, and TH12 with a left Hermite index of ^0 are zero. For similar reasons, the only possibility for the vertex on the right is Qc111 or Qc011. Thus we have

M(R,t;R′,t′)=∑λλ′∫dR′′dR′′′Qc111(R,^0;R′′,λ)

×χP(R′′,λ,t;R′′′,λ′,t′)

×(Qc111(R′′′,λ′;R′,^0)+Qc011(R′′′,λ′;R′,^0))

(12)

In a diagram for M, between the Q on the left and the Q on the right is a member of a series of diagrams that defines what shall be called the projected propagator χP. See the next subsection for the formal definition. This result implies that the time dependence of the memory function associated with the intermediate scattering function can be calculated from the time dependence of the projected propagator χP. (In Mori’s theory of memory functions,HMori () the time dependence of a memory function of a dynamical variable is given by an operator of the form exp(i(I−P)L) that acts on the vector space of all functions in classical N-particle phase space. Here L is the Liouville operator, P is a projection operator, and I is the identity operator. There is a close analogy to the results here, which is why we use the term ‘projected propagator’ for χP.)

Note that within the square brackets, the multiplications are Hermite matrix multiplications.

v.2 The projected propagator

Eq. (12) follows from the diagrammatic series for M in Sec. V.1.3 provided we define the projected propagator χP in the following way.

χP(R,λ,t;R′,λ′,t′)≡ the sum of all diagrams in the latest series for χ(R,λ,t;R′,λ′,t′) that have no χ(0)^0 bond whose removal would disconnect the roots. □

The projected propagator is of central importance in the present theory. To obtain an explicit, useful statement of its diagrammatic series, we note the following.

1. In every diagram included in the series for χP, each root is attached to a χ(0)P bond rather than a χ(0)^0 bond.

2. An MH or TH matrix element is equal to zero unless there is at least one Hermite index on the left and one on the right that is not ^0. (See Appendix C.3.) It is permissible to include the requirement that, in every diagram, every MH and TH vertex has at least one χ(0)P bond attached on its left and one on its right, since this does not change the value of the series.

We thus have the following result.

χP(R,λ,t;R′,λ′,t′)= the sum of all topologically distinct matrix diagrams with:

(v) each MH and TH vertex has at least one χ(0)P bond attached to its left and at least one χ(0)P attached to its right. □

v.3 The propagators associated with the longitudinal and transverse current correlation functions

To use Eqs. (7) and (8) to calculate the longitudinal and transverse current correlation functions, we need expressions for the ^z^z and ^x^x matrix elements of χ.
Consider the graphical expression for χ(R,λ,t;R′,λ,t′), for λ=^z or ^x, obtained by using the equation for χ in Sec. V.1.2.

We perform an analysis similar to that in Sec. V.1.3 for each of these two series. We examine each graph and find the χ(0)^0 bonds whose removal would disconnect the roots. Each graph has either no such bonds or one or more such bonds.

A diagram in the series with no such bonds is a member of the series for χP, and so the sum of these diagrams is χP(R,λ,t;R′,λ,t′).

If a graph has one or more such bond, we locate the leftmost and rightmost bond of this type. (They are the same bond in the case that there is only one.)

The part of the diagram that has these bonds and what is in between them (if there are two bonds) or the bond itself (if there is only one) is clearly a member of the graphical series for χ in which the bond attached to each root is a χ(0)^0 bond. The sum of all these possibilities is clearly equal to the ^0^0 element of χ itself.

The vertex to the immediate left of the bond on the left must be a Qc111 or Qc011, vertex because these are the only vertices with one right point that have a nonzero matrix element when the right index is ^0. The part of the diagram between the left root and this vertex is a member of the series for χP.

Similarly, the vertex to the immediate right of the bond on the right of the contribution to χ must be a Qc111, and what is between it and the right root is a member of the series for χP.

We have the following exact result, which is easier to represent in the Fourier-Laplace domain.

^~χ(q^k,z)λλ=^~χP(q^k,z)λλ

(14)

+[^~χP(q^k,z)(^Qc111(q^k)+^Qc011(q^k))]λ^0^~χ(q^k,z)^0^0

×[^Qc111(q^k)^~χP(q^k,z)]^0λfor λ=^z or ^x

v.4 Summary of this section

We stated the graphical expression for the propagator χ for density fluctuations in single particle phase space that incorporated the changes made in Secs. III and IV. These included approximating the short ranged repulsive parts of the Q vertices using a hard sphere model and representing the momentum dependence of diagrams in terms of Hermite matrix elements.

We showed that the χ^0^0, χ^z^z, and χ^x^x Hermite matrix elements of the propagator are directly related to the coherent intermediate scattering function, the longitudinal current correlation function, and the transverse current correlation function, respectively.

We defined a projected propagator χP.

We found that the memory function for the kinetic equation for χ^0^0 can be expressed simply in terms of χP, and therefore χ^0^0 can be calculated from χP.

We found that χ^z^z and χ^x^x can be expressed simply in terms of χP and χ^0^0.

It follows that a theory for the projected propagator χP can lead to theoretical calculations of the correlation functions mentioned.

vi.1 The generalized Enskog projected propagator

An important approximation for χP, and the starting point for our analysis, is the generalized Enskog projected propagator, which can be defined in the following way.

χEP(R,λ,t;R′,λ′,t′)≡ the sum of all topologically distinct matrix diagrams in the series in Sec. V.2 for χP(R,λ,t;R′,λ′,t′) that have no QL or TH vertices. □

Every diagram in this series is an alternating sequence of bonds and vertices that have one left point and one right point. Once this is recognized, it is clear that no diagram in the series has a χ(0)^0 bond, since that would violate the topological restriction (iii) in the diagrammatic expression for χP. Finally, no diagram can have a Qc011 vertex because such a vertex must be connected to one χ(0)^0 bond if its matrix element is nonzero. Thus we have the following.

χEP(R,λ,t;R′,λ′,t′)= the sum of all topologically distinct matrix diagrams with

(i) a left root labeled (R,λ,t) and a right root labeled (R′,λ′,t′);

(ii) free points;

(iii)χ(0)P bonds;

(iv)Qc111 and MH vertices;

such that

(i) the left root is attached to a bond and the right root is attached to a bond;

(ii) each free point is attached to a bond and a vertex. □

χEsP(R,λ,t;R′,λ′,t′)= the χEP(R,λ,t;R′,λ′,t′) series just above with MH vertices replaced by MHs vertices. □

If χEP is used as an approximation for χP to calculate the memory function of the intermediate scattering function in Eq. (12), the result is equivalent to the memory function of the generalized Enskog theory,BoonYip () as can be verified by a detailed calculation. This projected propagator contains the physics of uncorrelated binary hard sphere collisions, described by the hard sphere memory function MH, with a collision frequency ν that is proportional to the particle density and the radial distribution function at contact. (See Appendix B.5.) Some of the properties of χEP are discussed below in Sec. VII.2.

vi.2 Topological reduction of the series for χP

The series for χEP consists of simple chain diagrams containing n(≥1)χ(0)P bonds separated by n−1Qc111 and MH vertices.
It is straightforward to perform a topological reduction of the series for χP that eliminates these chains and replaces them by χEP bonds. This topological reduction eliminates all χ(0)P bonds and MH vertices, as well as all Qc111 vertices that are not attached to at least one χ(0)^0 bond.
Thus we have the following result.

χP(R,λ,t;R′,λ′,t′)= the sum of all topologically distinct matrix diagrams with:

Figure 2: Several graphs in the series of Sec. VI.2 for χP. Each small open circle is a root. Each small closed circle is a free point. Each large open circle is a vertex. The letter T indicates a TH vertex. A vertex without such a letter is a Qc11 or QcL vertex. Each solid line between vertices is a χ(0)^0 bond. Each wavy line is a χEP bond. Some vertices have internal lines. (Each maximal qs subgraph is surrounded by a rectangle of red dashed lines, which is not part of the diagram. Maximal qs subgraphs are discussed in Sec. VII.3.3.)

shows examples of the graphs in this series. This will be used as the starting point for the discussion of the overdamped limit in the next section.

vii.1 Mh, Th21, and the hard sphere collision frequency ν

The hard sphere memory function MH that appears in the present theory has appeared in previous kinetic theories of hard spheres, both in the context of the Boltzmann kinetic equation for the dilute gas phase and in the generalized Enskog theory for the dense liquid phase.MazenkoWeiYip (); MazenkoYip2 (); BoonYip () It describes the effect of one binary hard sphere collision on the momentum of a fluctuation. Appendix B.6 gives a formula for the MH function in the momentum representation, and Appendix C.3.2 gives an integral representation of the Hermite matrix elements of the function. All of the results for the hard sphere fluid obtained from the present theory appear to be equivalent to those obtained by Furtado et al.MazenkoYip2 (); noteFurtado ()

Any matrix element of MH or MHs with its left and/or right Hermite index equal to ^0 is zero. An explicit formula for the ^z^z element of MHs, given in Appendix B.6, reduces to the following.

^MHs(q)^z^z=−ν

The quantity ν has units of (time)−1 and is positive and independent of wave vector. Except for a numerical factor of approximately unity, it is equal to the hard sphere collision frequency according to the generalized Enskog theory. Also, except for a similar numerical factor, it is equal to the reciprocal of the relaxation time associated with the decay of the velocity autocorrelation function according to that theory. For simplicity, we shall refer to it as the generalized Enskog collision frequency.

In units in which the hard sphere diameter is approximately 1 and the mean thermal velocity is approximately 1, ν is a frequency large compared with 1, and ν−1 is a time small compared with 1 for atomic fluids at liquid densities. This is because the mean free path between hard sphere collisions of a single particle is much smaller than the hard sphere diameter for liquid densities.

The frequency ν enters the diagrammatic theory via the Qc1R21 vertex, which describes the rate at which a propagating density fluctuation collides with an equilibrium distribution of particles. This vertex contains a factor of ρgm (see Eq. (32)). In the limit that the repulsive potential becomes a hard sphere potential, gm becomes the pair correlation function at contact for the hard spheres. (See Appendix B.5.) This factor is the essential ingredient in ν. Both TH21 and MH contain one Qc1R21 vertex and hence are O(ν). All other hard sphere vertices are independent of the density and the pair correlation function and contain only essentially geometric information about the collisions of two hard spheres.

The diagrams in the series for χP in Sec. VI.2 contain TH21 vertices but no MH vertices. The latter have been absorbed into the χEP bonds that also appear in the series. The TH21 vertices generate positive powers of ν in the value of a diagram. As we shall see below, the χEP bonds lead to negative powers of ν. In the rest of this section, we explore the consequences of this and identify those diagrams that are most important for determining the behavior of χP for long times when ν is large.

vii.2 Properties of χEP

The χEP function represents the effect of a sequence of zero of more uncorrelated binary collisions on a density fluctuation, each of which is described by the